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A New Statistical Similarity Measure for Change Detection in Multitemporal SAR Images and Its
Extension to Multiscale Change Analysis
Jordi Inglada, Grégoire Mercier
To cite this version:
Jordi Inglada, Grégoire Mercier. A New Statistical Similarity Measure for Change Detection in Multitemporal SAR Images and Its Extension to Multiscale Change Analysis. IEEE Transactions on Geoscience and Remote Sensing, Institute of Electrical and Electronics Engineers, 2007, 45 (5), pp.1432-1445. �hal-00582539�
A New Statistical Similarity Measure for Change Detection in Multitemporal SAR Images and its
Extension to Multiscale Change Analysis
Jordi INGLADA, Gr´egoire MERCIER
Abstract— In this paper, we present a new similarity measure for automatic change detection in multitemporal SAR images.
This measure is based on the evolution of the local statistics of the image between two dates. The local statistics are estimated by using a cumulant-based series expansion which approximates the probability density functions in the neighborhood of each pixel in the image. The degree of evolution of the local statistics is measured using the Kullback-Leibler divergence. An analytical expression for this detector is given, allowing a simple compu- tation which depends on the 4 first statistical moments of the pixels inside the analysis window only.
The proposed change indicator is compared to the classical mean ratio detector and also to other model-based approaches.
Tests on simulated and real data show that our detector outper- forms all the others.
The fast computation of the proposed detector allows a multiscale approach in change detection for operational use.
The so-called multiscale change profile (MCP) is introduced to yield change information on a wide range of scales and to better characterize the appropriate scale. Two simple yet useful examples of applications show that the MCP allows the design of change indicators which provide better results than a monoscale analysis.
Index Terms— Change detection; multitemporal SAR images;
Kullback-Leibler divergence; Edgeworth series expansion; mul- tiscale change profile (MCP).
I. INTRODUCTION
REMOTE sensing imagery is a precious tool for rapid mapping applications. In this context, one of the main uses of remote sensing is the detection of changes occurring after a natural or anthropic disaster. Since they are abrupt and seldom predictable, these events cannot be well temporaly sampled – in the Shannon sense – by the polar orbit satellites which provide the medium, high and very high resolution imagery needed for an accurate analysis of the land cover.
Therefore, rapid mapping is often produced by detecting the changes between an acquisition after the event and available archive data.
This change detection procedure is made difficult due to the time constraints imposed by the emergency context. Indeed, the first available acquisition after the event has to be used whatever its modality which is more likely to be a radar image due to weather and daylight constraints.
The kind of changes produced by the event of interest are often difficult to model. The same kind of event – a flood – can have different signatures depending on where it happens – high density built-up areas, agricultural areas, etc. – and on the characteristics of the sensor. Also, the changes of interest are
all mixed up withnormal changes, which can be the majority if the time gap between the two acquisitions is too long.
All these issues present us with a very difficult problem:
detecting abrupt unmodeled transitions in a temporal series with only two dates1.
From this position of the problem, one can make the straightforward deduction that pixel-wise comparison between the two images will not be robust enough.
In the case of radar acquisitions, the standard detector is based on the ratio of local means [3]. This detector is robust to speckle noise, but it is limited to the comparison of first order statistics. The classical model for SAR intensity introduced by Ulaby et al. [4] assumes that the texture is a zero-mean multiplicative contribution. Therefore, changes taking place at the texture level which preserve the mean value will not be detected by the mean ratio detector. One can thus assume a miss-detection behavior of the detectors using only the mean pixel values. This remark invites a more accurate analysis of the local statistics of the images to be compared. Bujor et al. [5] did very interesting work by analyzing the interest of higher order statistics for change detection in SAR images.
They concluded that the ratio of means was useful for step changes and that the second and third order log-cumulants were useful for progressive changes appearing in consecutive images in multi-temporal series. Since higher order statistics seem to be helpful, one may want to compare the local probability density functions (pdfs) of the neighborhood of the homologous pixels of the pair of images used for the change detection.
Of course, this assumes that the pdfs are known, and that there exists a robust way to compare them. The estimation of pdfs can be made with different approaches, but the straight- forward histogram method should be avoided due to the need a high number of samples for the estimation. Indeed, small analysis window sizes are required to yield high resolution change maps. In this paper, we will present several approaches for this estimation by using only a small number of samples for the local statistics estimation, up to order 4.
Once the pdfs are estimated, their comparison can also be performed using different criteria. Information theory shows that a good measure is the Kullback-Leibler divergence, also called information gain. We will use a symmetrical version of this measure and show that it is superior to the classical
1In the case where a sequence of several images is to be processed, the approaches presented in [1], [2] may be applied.
detector when the pdfs are correctly estimated.
Therefore, these measures will be based on the comparison of local neighborhoods where an analysis window for the computation of the local estimation of probabilities is used.
The problem which arises here is the one of choice of window size. Since we are facing unmodeled changes, we cannot choose the window size to fit the size of the expected changes.
An inappropriate window size can produce miss- and over- detections:
• when using a small window for a correlation analysis, no detection will be performed in a homogeneous area, which was globally changed to another homogeneous area;
• on the contrary, when using a larger window size, change areas have to be of larger size or strong in intensity (relatively to the measure) to be detected. In these cases, it will produce a coarse resolution change map.
One way to overcome this problem is by applying a multiscale change detection analysis.
Scale is to be understood in its geographic meaning, which is the spatial extent of the study area. It does not refer to the cartographic meaning of scale (the larger the scale the more detailed the information) [6]. For an interesting discussion on scale issues in remote sensing see [7].
Image processing techniques for multiscale analysis often use the cartographic meaning and apply low-pass filtering and possibly sub-sampling. For change detection analysis, this filtering and sub-sampling can be justified in the case where the images are not perfectly registered [8]. In other cases we think that it is better to use all the available information, that is, maximizing the number of available samples by using increasing window sizes. Nevertheless, pyramidal multiscale decompositions can also be useful in the case of phenomena characterizations (see [9] for example).
Therefore, the main point of the problem is how to choose the largest window size which robustly detects the changes but which is small enough to preserve the resolution of the final map without miss-detections.
We propose to use multiscale change profiles, which are defined as the change indicator for each pixel in the image as a function of the analyzing window size. The computation of the change detection for each window size can be very time consuming. We present here a method for the computation of these profiles which allows the change indicator at scale n to be computed from the value obtained at scale n−1 plus a correction term which takes into account the addition of new samples only. Analytical expressions are given for three different change indicators.
This paper proposes three main contributions:
1) an Information Theory-based similarity measure which uses full local statistics;
2) the use of cumulant-based series expansions of similarity measures, which allow a robust and fast computation by using a small number of samples;
3) the concept of multiscale change profile and its fast implementation using recurrence evaluations.
The paper is organized as follows: section II presents the problem formulation; section III introduces the measures used
for the production of a change image; in section IV we intro- duce the concept of multiscale change profile and present the mathematical formulation allowing its optimized computation;
sections V and VI present the results obtained on simulated and real data respectively, and section VII concludes the paper and proposes some directions for future work.
II. PROBLEM FORMULATION
Let us consider two co-registered SAR intensity imagesIX andIY acquired at two different datestX andtY respectively.
Our objective is to produce a map representing the changes occurring in the scene between tX and tY. The final goal of a change detection analysis is to produce a binary map corresponding to the two classes:changeandno change. The problem can be decomposed into two steps: the generation of a change image and the thresholding of the change image in order to produce the binary change map. Figure 1 shows a block diagram describing a classical change detection pro- cessing chain.
Window
Size Window
Size
Ix Iy
Thresholding
Change/No change Change Indicator Sliding
Window (i,j)
Sliding Window (i,j)
Fig. 1. Block diagram for a classical change detection processing chain.
The overall performance of the detection system will depend on both the quality of the change image and the quality of the thresholding. In this work, we choose to focus on the generation of an indicator of change for each pixel in the image. For interesting approaches in the field of unsupervised change image thresholding, the reader can refer to the works of Bruzzone and Fern´andez Prieto [10], [11], Bruzzone and Serpico [12] and Bazi et al. [13]. The reader may note that some of these approaches need a statistical modelling of the detectors’ response, which is not presented here.
The change indicator can also be useful by itself. Indeed, often the end user of a change map wants not only the binary information, given after thresholding, but also an indicator of the intensity of the change and eventually a confidence level. In order to evaluate the quality of a change image independently of the choice of the thresholding algorithm, the evolution of the detection probability,Pdet as a function of the false alarm
probability, Pfa, may be evaluated in the case where a set of constant thresholds are applied to the whole image. These are the so-called Receiver Operating Characteristics (ROC) and the plots of Pdet(Pfa)are called the ROC plots.
III. DISTANCE BETWEEN PROBABILITY DENSITIES
The main difficulty in the multitemporal analysis of SAR images is the presence of speckle noise. When moving away from interferometric configurations, the speckle is different from one image to the other and it can induce a high number of false alarms in the change detection procedure. Because of the multiplicative nature of speckle, the classical approach in SAR remote sensing involves using the ratio of the local means in the neighborhood of each pair of co-located pixels.
The Mean Ratio Detector, MRD, is usually implemented as the following normalized quantity:
rMRD=1−min µX
µY,µY µX
, (1)
whereµX andµY stand for the local mean values of the images before and after the event of interest. The logarithm of eq. (1) may also be used. Nevertheless, this operation does not modify the performance of the detector in terms of ROC even if the contrast of the image of change indicator is modified.
However, the logarithm is used since it modifies the initial pdf of the image of change indicator and then facilitates the development of Bayesian thresholding approaches [13].
This detector assumes that a change in the scene will appear as a modification of the local mean value of the image. If the change preserves the mean value but modifies the local texture, it will not be detected.
The change detection algorithm proposed in this paper ex- tends the MRD by analyzing the modification of the statistics of each pixel’s neighborhood between the two acquisition dates. A pixel will be considered as having changed if its statistical distribution changes from one image to the other.
In order to quantify this change, a measure, which maps the two estimated statistical distributions (one for each date at a co-located area) into a scalar change index is required. Several approaches could be taken into consideration: the mean square error between the two distributions, the norm of a vector of moments, etc. We have chosen to use a measure derived from Information Theory called the Kullback-Leibler divergence [14].
A. Kullback-Leibler divergence
Let PX and PY be two probability laws of the random variables X andY. The Kullback-Leibler divergence fromY toX, in the case where these two laws have the densities fX and fY, is given by :
K(Y|X) = Z
logfX(x)
fY(x)fX(x)dx. (2) The measure log ffX(x)
Y(x) can be thought of as the information on x for discrimination between hypothesis HX and HY if hypothesis HX is associated with pdf fX(x), and HY with fY(x). Therefore, the Kullback-Leibler divergenceK(Y|X)can
be understood as the mean information for discrimination betweenHX andHY per observation. This divergence appears to be an appropriate tool to detect changes when we consider that changes on the ground induce different shapes of the local pdf.
Since the Kullback-Leibler divergence can be understood as the entropy of PX relative to PY, it is also calledinformation gain. It can easily be proved thatK(Y|X)>0;K(Y|X)vanishes only when the two laws are identical.K(Y|X)can be used as a measure of thedivergencefromPY toPX. This measure is not symmetric as it stands: K(Y|X)6=K(X|Y), but a symmetric version may be defined by writing:
D(X,Y) =D(Y,X) =K(Y|X) +K(X|Y) (3) that will be called the Kullback-Leiblerdistance(KLD).
In order to estimate the KLD, the pdfs of the two variables to be compared have to be known. As stated in the introduction, the processing of high resolution change maps requires anal- ysis windows of small size, which makes impossible the use of local histogram estimations. In the following sections, we will introduce several approaches which allow the estimation of the pdfs by using a limited number of samples only. This requires some a prioriinformation on the data which can be introduced by using models of local statistics.
B. Gaussian KLD
As seen in section III, the classical detector of eq. (1) uses first order statistics only. Yet, second order statistics are often used for SAR image processing. For instance, many speckle reduction filters [15], [16], [17] are based on the contrast coefficientσ2X/µ2X, that is, the ratio between the variance and the square of the mean value. If the local statistics have to be compared up to the second order, the local random variables, X andY may be assumed to be normally distributed (i.e. of Gaussian law). Then, the pdf ofPX can be written as:
fX(x) =G(x;µX,σX) = 1 q
2πσ2X e
−(x−µX)2
2σ2X . (4)
An analogous expression holds for fY(x).
Fig. 2(b) shows the Gaussian approximation of the prob- ability distribution of a small region of interest (Fig. 2(a)) extracted from a SAR image.
If this Gaussian model is used in eq. (3), it yields the Gaussian Kullback-Leibler detector (GKLD):
rGKLD=σ4X+σ4Y+ (µX−µY)2(σ2X+σY2)
2σ2Xσ2Y −1. (5) It can be seen that even in the case of identical mean values, this detector is able to underline the shading of textures which is linked to the local variance evolution.
Nevertheless, the reader should note that the Gaussian model should not be used since SAR intensity values are always positive. However, this example has been given as a simple case of a parametric model which takes into ac- count second order statistics. Since some Gaussianity may be introduced into the data when resampling and filtering the
images during the pre-processing step, the Gaussian model may nevertheless be justified.
C. KLD using the Pearson system
The drawback of the GKLD is that SAR intensity statistics are not normally distributed, and the use of a bad model can induce bad performance of the detector, whatever the accuracy of the parameter estimation. In the absence of texture, the radar intensity follows a Gamma distribution:
fX(x) = 1 Γ(L)
L σX
L
e
−Lx
σXxL−1. (6) The Gamma distribution is characterized by the following parameters:L is the number of looks andσX, the square-root of the intensity SAR image. Γ(·)is the Gamma function.
In the presence of texture, the local statistics can deviate from the Gamma distribution. For instance, if the texture is modeled by a Gamma distribution with a shape parameter ν, the resulting intensity distribution follows a K-law [18]:
fX(x) =2 x
Lνx µX
L+ν
1
Γ(L)Γ(ν)Kν−L 2 Lνx
µX 1/2!
, (7)
whereK(·)is the modified Bessel function of the second kind andµX is the mean ofX.
More generally, it is now accepted that the statistics of SAR images can be well modeled by the family of probability distributions known as the Pearson system [19]. It is composed of eight types of distributions among which the Gaussian and the Gamma distributions may be found. The Pearson system is very easy to use since the type of distribution can be inferred from the following parameters:
βX;1=µ2X;3
µ3X;2 and βX;2=µX;4 µ2X;2,
where µX;i is the centered moment of order i of variable X.
That means that any distribution from the Pearson system can be assessed from a given set of samples by computing the first 4 statistical moments. Any distribution can therefore be represented by a point on the (βX;1,βX;2)plane. For instance, the Gaussian distribution is located at (βX;1,βX;2) = (0,3), and the Gamma distributions lie on theβX;2=32βX;1+3 line.
Details about the theory of the Pearson system can be found in [20].
Figure 2(c) shows an example of distribution estimation.
The Pearson approximation fits the data better than the Gaus- sian one (Fig. 2(b)). The example shown corresponds to a Beta distribution of the first type with parametersβ1=2.51×10−6, β2=1.87.
The Pearson-based Kullback-Leibler Detector (PKLD) was originally introduced in [21]. It does not have a unique analytic expression, since 8 different types of distribution may be hold.
Therefore, 64 different possibilities for the couples of pdf exist.
Once the couple of pdfs is identified, the detection can be
performed by numerical integration:
rPKLD(X,Y) = Z
log
fX(x;βX;1,βX;2) fY(x;βY;1,βY;2)
fX(x;βX;1,βX;2) +log
fY(x;βY;1,βY;2) fX(x;βX;1,βX;2)
fY(x;βY;1,βY;2)
dx.
(8) The correct way to proceed to use the Pearson system is to choose a pdf using the estimated moments and then estimate the parameters of the distribution by maximum likelihood.
While this can improve the results of the pdf estimation, the effect is not noticeable in terms of the estimation of the change indicator. This approach was not used in the experiments in order to reduce the computation cost.
The reader should note that in the case of single-look high resolution SAR data (better than 10 m) other statistical models may be more appropriate, mainly on urban areas. Nicolas et al. have proposed a new model based on the log-statistics and a set of pdfs coming from the Fisher system of distributions [22], [23]. It has been applied to high resolution SAR images on dense urban areas with promizing results [24], [25].
D. Cumulant-based KL approximation
Instead of considering a parameterization of a given density, or set of densities, it may be of interest todescribethe shape of the distribution. Such a description is based on quantitative terms that may approximate the pdf itself. The cumulants themselves do not provide such a pdf estimation directly but are necessary to describe its shape: for example, third order (κ3) is linked to the symmetry (i.e. skewness), while the fourth (κ4) is linked to the flatness (i.e.kurtosis). The density is then estimated through a series expansion. In fact, the cumulant generating function is used for such an estimation.
By definition, the cumulant generating function KX(·) of a random variableX is defined by:
KX(ω) =lnMX(ω) =∑
n
κX;n
ωn n!
withMX(·)being the moment generating function defined by:
MX(ω) = Z
eωxfX(x)dx= Z
1+ωx+ω22x2+· · ·
fX(x)dx.
For the case of the four first order cumulants, the following expressions hold [26, p.8]:
κX;1=µX;1 κX;2=µX;2−µ2X;1
κX;3=µX;3−3µX;2µX;1+2µ3X;1
κX;4=µX;4−4µX;3µX;1−3µ2X;2+12µX;2µ2X;1−6µ4X;1. (9)
Let us assume that the density to be approximated is nottoo far [27] from a Gaussian pdf (denoted asGX to underline the fact that it has the same mean and variance asX), that is, with a shape similar to the Gaussian distribution. The difference between KX(·) and KGX(·), can be written in terms of the
difference of the cumulants κX;n−κGX;n. By inversion, the density may be expressed by a formal Taylor-like series:
fX(x) =GX(x) +c1dGX
dx +c2d2GX
dx2 +· · · Since a Gaussian density is used, it yields
fX(x) =
∞ r=0∑
crHr(x)GX(x),
with Hr(x) known as the Chebyshev-Hermite polynomial of order r [27]. When choosing a Gaussian law so that its first and second cumulants agree with those of X, the number of terms of the series expansion is greatly reduced. This is the so-called Edgeworth series expansion. Its expression, when truncated to order 6, is the following:
fX(x) =
1+κX′;3
6 H3(x) +κX′;4
24 H4(x) +κX′;5
120H5(x) +κX′;6+10κ2X′;3
720 H6(x)
! GX(x).
(10)
It can be thought of as a model of the form X =XG+X′ where XG is a random variable with Gaussian density with same mean and variance asX, andX′, a standardized version of X [28] with:
X′= (X−κX;1)κ−1/2X;2 .
Fig. 2(d) shows an example of such an approximation of a histogram.
The Edgeworth series expansion of the two pdfs fX and fY may be introduced into the Kullback-Leibler divergence (eq. (2)). It yields an approximation of the Kullback-Leibler divergence by Edgeworth series, truncated at a given order.
In [29], such an approximation has been truncated to order 4 by using the equality ffX
Y =GfX
X
GX GY
GY
fY, whereGX (resp.GY) is a Gaussian density of same mean and variance as fX (resp.
fY). Then,
KLEdgeworth(X,Y) = 1
12 κ2X′;3
κ2X;2 +1 2
logκY;2
κX;2
−1+ 1 κY;2
κX;1−κY;1+κ1/2X;22
− κY′;3
a1 6 +κY′;4
a2 24+κ2Y′;3
a3 72
−1 2
κY2′;3
36 c6−6 c4 κX;2
+9 c2 κY2;2
!
−10κX′;3κY′;3(κX;1−κY;1) (κX;2−κY;2)
κ6Y;2 (11)
where
a1=c3−3 α κY;2
a2=c4−6 c2 κY;2
+ 3 κY2;2 a3=c6−15 c4
κY;2
+45 c2 κ2Y;2− 15
κ3Y;2 c2=α2+β2
c3=α3+3αβ2 c4=α4+6α2β2+3β4
c6=α6+15α4β2+45α2β4+15β6 α=κX;1−κY;1
κY;2
β=κ1/2X;2 κY;2
.
Finally, the cumulant-based Kullback-Leibler detector (CKLD) between two observationsX andY is written as:
rCKLD=KLEdgeworth(X,Y) +KLEdgeworth(Y,X). (12) The reader should note the fact that, like for the Pearson-based detector, despite the apparent complexity of the formulas, and thanks to eq. (9), only the moments up to order 4 have to be computed.
IV. MULTISCALE CHANGE PROFILE
Scale plays a strategic role in image analysis and more especially in change detection applications. In section I, it has been shown how an inappropriate scale of analysis can produce miss- or over-detections. In [30], Bovolo and Bruzzone, stress the fact that the scale of analysis is a key parameter for better discrimination between change andno change areas. Such a point of view is implemented by a wavelet transform of the log-ratio estimated with a window of a user-defined size.
Instead of applying a multiscale analysis of the change image, the purpose here is to produce a set of change indicators estimated at various scales. We will call itmultiscale change profile (MCP).
As stated in the introduction, the multiscale term refers here to the size of the analyzing window. The MCP will therefore involve computing the change indicator for a pixel by using neighborhoods of increasing sizes. The so-called profile corresponds to the sequence of change measures as a function of scale. We will restrict our formulation to the case of the CKLD. Given the fact that this detector needs the estimation of the statistical moments of the samples inside the analyzing window, we are interested in finding an approach which avoids the computation from scratch of the moments at every scale.
A. Optimized computation of the MCP
Let us consider the following problem: how to update the moments when anN+1th observationxN+1is added to a set of N observations {x1,x2, . . . ,xN} already processed. When
(a) ROI extracted from a SAR image
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
0 50 100 150 200 250 300
Probability
Value Gaussian approximation
Histogram Gaussian pdf
(b) Gaussian fitting
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
0 50 100 150 200 250 300
Probability
Value Pearson approximation
Histogram Pearson pdf
(c) Pearson fitting:β1=2.51×10−6,β2=1.87.
0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
0 50 100 150 200 250 300
Probability
Value Edgeworth approximation
Histogram Edgeworth pdf
(d) Edgeworth approximation
Fig. 2. Approximation of a histogram, coming from a 50×50 Region of Interest (ROI), using three different strategies. The Pearson fitting yields a Beta distribution of the first type.
considering raw moments of order r, the formulation comes easily as:
˜
µr,[N+1]= N
N+1µ˜r,[N]+ 1 N+1xrN+1.
˜
µr,[N] (resp. ˜µr,[N+1]) stands for the raw moment of order r estimated with N samples (resp. N+1 samples). Since the analyzing window may contain textured areas, the mean value itself may be modified by the increase in the number of samples. Therefore, by using simple binomial properties, it can be shown that central moments may be characterized by:
µ1,[N]= 1
Ns1,[N] (13)
µr,[N]= 1 N
r ℓ=0∑
r ℓ
−µ1,[N]r−ℓ
sℓ,[N], where the notation sr,[N]=∑Ni=1xri has been used.
Hence, when considering a new samplexN+1, each moment may be updated directly by using updates ofs1,[N+1] and then sr,[N+1]for increasing values of orderr. The Edgeworth series is also updated by transforming moments to cumulants (by using eq. (9)) to be introduced in eq. (10) and then in eq. (11).
Fig. 3 shows an example of a pdf estimation on a homo- geneous area (shown in fig. 2(a)) when the window increases from 9×9 to 17×17. In fact, the availability of updating the estimation of the distance between distributions from windows of any size without re-processing the overall data is the most
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
0 50 100 150 200 250 300
Probability
value Edgeworth Approximation
Histopgram window 9x9 window 13x13 window 17x17
Fig. 3. Example of pdf estimation update by increasing sample set from a window of size 9×9 to 17×17. The histogram has been estimated with a 17×17 window.
interesting point for multiscale change detection purposes.
This on-line multiscale moment estimation is the key for the operational use of the MCP concept.
For example, the computation of rCKLD with windows of size ranging from 5×5 pixels to 51×51 pixels (22 different window sizes) takes only 42% additional time with respect to the computation of a single detection with a window of median size of 29×29 pixels (300 s. versus 210 s. for a 800×400 pixel image).
B. MCP exploitation
The MCP computation produces a multichannel image (one scale per channel) whose pixels have to be transformed into scalar values in order to provide a change indicator. In order to exploit the information available at all scales, two approaches may be investigated. The first one consists in choosing thebest scale for each image pixel. The second one consists infusing the information available at all scales in order to provide a single change value.
The development of an optimal approach for the exploitation of the MCP may be application dependent. Indeed, multiscale fusion approaches could be tuned to a particular type of change – shape, nature, etc. In this section, two simple yet useful choices will be proposed which yield an improvement in comparison to the performance of a single scale detection.
1) In order to choose thebestscale, we will choose the one which produces the highest KLD value. This assumes that this scale is the one that is associated with the largest window inside a homogeneous area with respect to the classes changeandno change.
2) The fusion of the multiscale information will be per- formed by using Principal Component Analysis (PCA).
The first principal component of the MCP multichannel image will be considered as the change indicator. This corresponds to a linear combination of all scales which maximizes the contrast of the final image.
V. EXPERIMENTS WITH SIMULATED DATA
A. Data set description
Simualtions have been performed to better understand the behavior of the detectors relatively to a given kind of change and a given size of the change area. Since this study focuses on change detection on radar images, a speckle simulation is performed from a map of ground reflectivity. The simulated changes are applied on a small area, drawn as a circle, located in the center of the initial image.
The simulation procedure is based on the radar image formation mechanism. Each pixel is simulated with a given amplitude (coming from a SPOT, NIR band image, normalized to [0,1]) and thousands phases coming from independent uniform generations in[0,2π[to characterize elementary wave scatterers. Taking the square of the modulus of each pixel yields a 1-look intensity image. A 4-look instensity image is obtained by averaging and subsampling two adjacent pixels along lines and rows.
Each simulation of change is applied to the initial image by using a change circle of given size taken from {5,10,15,20}.
Once the speckle simulation is performed (independently from one image to another), the speckled-changed images are mosaiked on a 2×2 grid as shown on fig. 4(b).
B. Simulation of changes
Three kinds of change were considered:
1) Offset change: fig. 4(c). The initial image is modified by applying an offset value (i.e.a shift) to the inital data. This is a very simple type of change which seldom occurs in reality, but is useful to characterize the behavior of the detectors.
(a) Before (b) Mask
(c) After Offset (d) After Gaussian
(e) After Deterministic Fig. 4. Simulated data set.
2) Gaussian change: fig. 4(d). The initial image is modified by applying a zero mean gaussian additive noise to the initial data. This corresponds to a change in the state of the surface – field, vegetation. This is the main type of change that one can encounter in medium resolution SAR images.
3) Deteministic change: fig. 4(e). The initial image is modified by pasting values copied from another area of the image itself. This type of change can occur when there is a land-use change, anthropic activities, etc.
C. Results
1) Mono-scale detection: the results of the different detec- tors for a fixed analysis window size are analyzed.
Figure 5 shows the ROC plots for the case where the change consists in a shift of the reflectivity value (fig. 4(c)). In this case, all 4 detectors are able to detect the changes with high accuracy. There is a slight difference in performance between the pair CKLD - GKLD and the pair PKLD - MRD, but it is difficult to infer general behavior from this result. To draw a preliminary conclusion, for a simple change such as a reflectivity shift, the mean value criterion is efficient enough for good discrimination in the changes, even on speckled images.
Figure 6 shows the ROC plots in the case of a Gaussian change. The change is simulated by the addition of a Gaussian noise to the reflectivity (before speckle simulation). In this
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Pfa ROC plots - Offset
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Fig. 5. ROC plot comparison of the 4 detectors for a simulated change consisting in an offset on reflectivity.
case, the mean value of the observed pixels remains approx- imately the same. It is difficult for this kind of change to be observed by a human operator. However, it is more likely to occur when modifications affect the surface without changing its nature. In this case, even if all the detectors show bad performance, in comparison to the offset case, the MRD and the PKLD are far below the GKLD and CKLD. The bad performance of the MRD is easy to understand, since the zero- mean Gaussian noise added to the reflectivity slightly changes the observed mean value. For the PKLD, it can be argued that the type of law in the Pearson system is not very different from the initial case and the main difference is seen through the mean value, thus obtaining the same performance as the MRD. On the contrary, the GKLD assumes a simpler model than the PKLD and is able to take into account the mean and the variance modifications together. Finally, the ability of the CKLD to fit many different types of densities, allows better detection for this difficult type of change.
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Pfa ROC plots - Noise
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Fig. 6. ROC plot comparison of the 4 detectors for a simulated change consisting in a Gaussian random modification of the reflectivity.
The third type of change is that of a texture change which can occur when there is a land use change, anthropic activities, etc. In this case, as can be infered from figure 7, the mean value of the regions may or may not change and it is therefore interesting to analyze the shape of the density. The Pearson detector can be even worse than the MRD when the model does not fit the data, which is the case in presence of mixtures.
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Pfa ROC plots - Texture
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Fig. 7. ROC plot comparison of the 4 detectors for a simulated change consisting in a deterministic modification of the reflectivity.
2) Analysis of the MCPs: some collected multiscale change profiles, obtained by applying rCKLD of eq. (12) to our data set are analyzed. Four different profiles are presented. They are extracted from a change area of the simulated data set for the case of a deterministic texture change and a radius of 10 pixels. These profiles are labeled as follows:Far for the case where the analysis window is located 30 pixels from the center of the change area;Outside borderfor a distance of 15 pixels;
Inside borderfor a distance of 7 pixels andInside centeredfor a distance of 0 pixel. Fig. 8(a) presents a diagram explaining how the profiles are extracted with respect to the change area and fig. 8(b) presents the profiles themselves.
The Far profile shows low values for small window sizes, and these values increase as the window size increases and it begins to include pixels from the change area. The values decrease for large window sizes, since the window stops including newchangepixels while includingno changepixels present in all directions. The Outside border profile has a similar behaviour, but the CKLD values are high for small scales since the pixel is nearer to the change area. TheInside borderprofile shows higher values for the change indicator for small window sizes. Finally, theInside centeredprofile shows very high values of the detector for a large interval of window sizes. It is worth noting that the CKLD values are nearly the same for all detectors for the largest window sizes, since, at this scale, all detectors include the same proportion ofchange andno changepixels.
3) MCP exploitation: in this section, the interest of the use of the MCP is illustrated with respect to the selection of a fixed scale of analysis, (i.e. a fixed window size). The MCP allows the best scale to be selected for each pixel location in the images. Here the maximum of the profile is used as a means to select the appropriate scale.
The maximum of the MCP and 2 different scales, 5×5 and 17×17 are compared. The small window size is used in order to detect small changes, but its main drawback is that the false alarms may increase in the presence of noise. The larger window size gives a lower false alarm rate since the noise is averaged, and therefore its effect is reduced. But small changes can also be averaged and therefore the detection probability
